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To solve this problem, we can set up a system of equations based on the amount of candy pieces in each mixture and the total amount of trail mix needed.
Let x be the number of cups of the 10% candy trail mix needed and y be the number of cups of the 25% candy trail mix needed.
We can then set up the following equations:
0.10x + 0.25y = 0.20(18)
x + y = 18
Solving this system of equations, we get x = 12 and y = 6.
Therefore, 12 cups of the 10% candy trail mix and 6 cups of the 25% candy trail mix would be needed to make 18 cups of trail mix that is 20% candy pieces.